An Introduction Into The SVAR Methodology: Identification .

5(8)mt γ 2 yt Bmy ( L) yt Bmm ( L)mt ems t ,where B( L) denotes polynomials in the lag operator L and Σ e is again thevariance-covariance matrix of the structural disturbances.10 The first equationshows the impact of the monetary policy instrument on real activity. Thisequation is interpreted as an aggregate demand relation parsimoniouslyspecified. An equation like (7) is often used to obtain estimates of the so-calleddynamic multipliers of monetary policy which describe the impact of themonetary policy instrument on output. The dynamic multipliers are useful, forexample, to determine the value to be assigned to m t to achieve a given path forthe macroeconomic variable yt .11 The second equation can be interpreted as amoney supply function. Here, we assume that the central bank sets the moneysupply according to a feedback mechanism involving current output and thehistory of both variables, while discretionary policy actions are captured by themoney supply shock ems .As discussed in the preceding section, there is no way to obtain estimates ofthe structural parameters of interest without some identifying restrictions. Thereduced form of (7) and (8) is given by the following set of equations,(9)y t B *yy ( L ) y t B *ym ( L ) mt ud t* ( L) y B * ( L ) y u(10) mt Bmytmmtms t ,where B * Γ 1B and u Γ 1e , as before. Assuming a uniform lag length of k itis apparent that the reduced form represented by (9) and (10) has 4k coefficientswhile the structural model represented by (7) and (8) has (4k 2) coefficients,so one identifying restriction for each equation is needed to obtain estimates ofthe structural parameters from the data.As noted above, identification in simultaneous equation models is typicallyachieved by imposing exclusion restrictions on the elements of the matrices Γand B . These restrictions are imposed on the model on a priori grounds and10 The lag polynomial B( L) takes the general form B(L ) b1L b 2 L2 . b n Ln .11 See Bagliano and Favero (1998), pp. 1071.

6cannot be tested. For this reason they should be based on a firm theoreticalfoundation.Regarding restrictions on Γ , one could argue that due to lags in the collectionof statistics on economic activity monetary policy makers cannot observe outputwithin the period, and, therefore, cannot respond contemporaneously to theoutput variable. This would suggest restricting the parameter γ 2 to zero. Onecould also argue that monetary policy affects output only with a delay due tolags in the transmission mechanism. According to this argument, the parameterγ 1 could be set to zero. With these two restrictions the matrix Γ becomes theidentity matrix and the reduced form given by (9) and (10) actually represents astructural model of the economy. For the moment, we will not pursuerestrictions on the simultaneous relationships between the variables further, butreturn to this issue in the context of the SVAR analysis where this type ofrestriction is very popular.The model can also be identified by imposing restrictions on the elements ofthe matrix B . The matrix B describes the effects of the lagged endogenousvariables on output and money. That is, this matrix describes the dynamicrelationships between the variables in the model. The lagged endogenousvariables are predetermined, meaning that they do not correlate with thecontemporaneous or future realizations of the structural shocks. Variables thatare predetermined can be treated, at least asymptotically, as if they wereexogenous.12 Even though this makes these variables easy to handle empirically,restrictions on lagged endogenous variables are difficult to justify from atheoretical perspective, since economic theory usually does not say muchregarding the dynamic relationships between variables, and for this reason it ispreferable to let these coefficients be determined by the data. 13 In SVARmodels, no restrictions are imposed on the elements of B .Another approach is to search for exogenous variables to help with identification. 14 A variable is defined as strongly exogenous if it does not correlate with12 See Greene (1997), p. 714.13 For a discussion see also Amisano and Giannini (1997), pp. 22.14 Inclusion of exogenous variables increases the chances for the model to be identified. SeeFavero (2001), pp. 88.

7the contemporaneous, future or past realizations of the structural shock in theequation. 15 This is a stronger condition than that holding for predeterminedvariables, but from the standpoint of identification both types of variables can betreated in a similar manner. 16 Since the use of exogenous variables foridentification is specific to dynamic simultaneous equation models in the sensethat SVAR models consist only of endogenous variables, we concentrate in thefollowing on the role of exogenous variables in the identification of our smallsimultaneous equation models. This will prove useful in bringing out thefundamental difference in identification between dynamic simultaneous equation models and SVAR models. As regards the structural model considered here,we need at least two exogenous variables to achieve identification. One of thosetwo variables should be highly correlated with the aggregate demand variablebut not with the policy instrument, whereas the opposite should hold for theother variable. In the following two subsections we illustrate how exogenousvariables which fulfill these requirements can help with the identification of themoney supply and the aggregate demand relations.1.Identification of the money supply scheduleTo illustrate the identification principle for the money supply relation, we makethe reasonable assumption that fiscal policy, which is exogenous to our model, isa major determinant of aggregate demand conditions, but is not a factor in thesetting of the monetary policy course. That is, we assume that this variable canbe restricted on a priori grounds to be irrelevant for the determination of moneysupply. Setting the coefficient for this variable to zero in the money supplyequation provides the identifying restriction needed to estimate the structuralparameters in this equation. The identification principle is illustrated with thehelp of the following diagram:15 See Hansen (1991), p. 340.16 See Greene (1997), pp. 714, and Favero (2001), pp. 88.

8Identifying the money supply scheduleFigure 1:mMSdG 2xCAD 2xxABdG 1AD 0AD 1yFigure 1 plots the money supply schedule MS and the aggregate demandschedule AD . Initially, the system is at point A . Next, fiscal policy is assumedto become expansionary, which is denoted by dG1 . According to the identifyingrestriction this change in the fiscal policy stance only shifts the aggregatedemand schedule, but not the money supply schedule. As regards this point,recall that the fiscal policy coefficients in the money supply function have beenset to zero, so that there is no direct response of the money supply to the fiscalpolicy stance. This restriction ensures that the money supply schedule is pinneddown in Figure 1 with respect to the fiscal policy stance. Following the fiscalimpulse, the system reaches a new equilibrium in B . Next, fiscal policy isassumed to become restrictive ( dG2 ), moving the system to C . To see how thisprocedure identifies the money supply equation, it is useful to notice that thepoints A , B and C provide a good description of the money supply scheduleMS . In other words, changes in the fiscal policy stance are an exogenous sourceof shifts in the aggregate demand schedule and help to trace out the MSschedule, which is being pinned down by the identifying restriction.With the help of the fiscal policy variable and the accompanying identifyingrestriction it also possible to use regression analysis methods like the two-stageleast square method to obtain consistent estimates of the structural parameters in

9the money supply equation. 17 Using an instrumental variables approach liketwo-stage least squares, the fiscal policy variable serves in the estimation ofequation (8) as an instrument variable for the contemporaneous output variable.For the discussion of this approach it is useful to reformulate the identificationproblem: If one estimates equation (8) using ordinary least squares (OLS), thiswould lead to an inconsistent estimate of the parameter γ 2 , because theresulting estimate would represent an average of the structural parameters γ 1 andγ 2 , with weights depending on the sizes of the variances of the structuraldisturbances ed and ems . This is known as simultaneous equation bias. 18Technically, this bias arises because for the contemporaneous output variable inequation (8) the condition is violated that the determining variable needs to beindependent of the disturbance term if the OLS estimator is to be consistent. 19The source of the problem is that the contemporaneous output variable is anendogenous variable and, therefore, it is correlated with the disturbance termems ,t . In other words, the OLS estimate of γ 2 is biased because output andmoney in our model are simultaneously determined and, hence, the outputvariable is a function of the disturbance term of the money supply equation. Theintuition behind the instrumental variables approach is that by using for theendogenous determining variable an instrument which is uncorrelated with thedisturbance term this approach reestablishes the orthogonality between thedetermining variable and the disturbance term, thereby obtaining a consistentestimator. 20In our case the instrumental variables approach requires a variable that ishighly correlated with the contemporaneous output variable, but uncorrelatedwith ems ,t . The fiscal policy variable is such an instrument. On the one hand,this variable is likely to be highly correlated with output because it is animportant factor for aggregate demand conditions. On the other hand, it isuncorrelated with the disturbance term ems ,t , because fiscal policy is assumed to17 See Hamilton (1994), pp. 238.18 See Hamilton (1994), p. 234.19 See Favero (2001), p. 107.20 For a detailed exposition of the instrumental variables estimator, see Favero (2001), pp.108.

10be an exogenous variable and, therefore, it is not a function of the money supplyvariable. 21 Finally, according to our identifying restriction fiscal policy is not adetermining variable in the money supply equation. If it were, it could notsimultaneously serve as an instrument for another determining variable in thisequation. In other words, the fiscal policy variable would not add a new sourceof information to our estimation problem in this case. But our identifyingrestriction rules this case out, thereby ensuring that the fiscal policy variable is avalid instrument.2.Identification of the aggregate demand scheduleFor the estimation of the structural parameters in the aggregate demand relationan instrument is needed that is correlated with the money supply variable but notwith the disturbance term ed ,t . Moreover, this variable should not be a factor indetermining aggregate demand. Finding such a variable poses a considerablechallenge. One candidate is the term spread. This variable is correlated withmoney supply if monetary policymakers accommodate shifts in money demanddue to portfolio reallocations, which are due to exogenous changes in the termspread.22 In addition, one has to assume that the term spread is exogenous withrespect to output, to ensure that it is not correlated with the disturbance term e d .That is, it is assumed that the term spread is not influenced by aggregate demandconditions. This is harder to justify; for instance, in an economic upswing thedemand for long-term capital typically rises, leading to higher long-term interestrates and thereby increasing the term spread.23 Finally, one has to assume thatthe term spread has no direct effect on aggregate demand, which represents ouridentifying restriction. This assumption is also hard to justify if agents areforward looking. We will return to this issue below. If all three assumptions21 If the exogeneity assumption does not hold the fiscal variable would be just anotherendogenous variable like output. In this case the model given by (7) and (8) should beextended by an additional equation modeling the fiscal policy stance as a function of thecontemporaneous monetary policy stance.22 The term spread is often used to model the opportunity costs of holding money. Changesin this variable lead therefore to changes in money demand. For an empirical model ofmoney demand with this specification, see for example Coenen and Vega (1999).23 For a discussion of the determinants of the yield spread, see Berk and Van Bergeijk(2000), pp. 5.

11hold, movements in the term spread shift the money supply function and thushelp to trace out the aggregate demand schedule, which remains fixed.Another common assumption for the estimation of the aggregate demandrelation is that the money variable in (7) is not an endogenous but an exogenousvariable. 24 With this assumption no identification problem arises in the firstplace. This allows us to estimate (7) in a straightforward way using ordinaryleast squares, because the problem of endogenous money is not an issueanymore. In terms of Figure 1 the money supply schedule is vertical. Thisassumption would hold, for example, if the central bank sets the money supplyaccording to some predetermined schedule (for example a k% rule). Thisassumption has an interesting but often unnoticed implication for the variancecovariance matrix of the structural disturbances, Σ e : Since money is exogenouswith respect to output, the coefficients in γ 2 and Bym ( L) in the money supplyequation are zero and, moreover, the money variable is uncorrelated with theaggregate demand disturbance ed . From this follows that the structural disturbances ed and ems are orthogonal. 25 This result will be of some significancein the comparison of identification in dynamic simultaneous equation modelsand SVAR models.III.Objections to the traditional approach to identification indynamic simultaneous equation modelsWhat, if any, are the problems with this approach to identification? A forcefulcritique comes from Sims (1980) who argues that truly exogenous variables arehard to come by. He notes that many exogenous variables in large macroeconomic models are treated as exogenous by default rather than as a result ofthere being a good reason to believe them to be strictly exogenous.26 Regardingpolicy variables, he points out that these typically have a substantial endogenous24 For a discussion, see Bagliano and Favero (1998), pp. 1071, and Sims et al. (1996), pp. 6.25 See also the discussion in Sims et al. (1996), pp. 6.26 See Sims (1980), p. 5.

12component, which precludes treating them as exogenous.27 Moreover, Simsargues that there are only a few powerful a priori identifying restrictions.28 Thisholds in particular when one allows for agents forming their decisions on thebasis of rational expectations and inter-temporal optimization. The textbookparadigm for identification is a simultaneous equation model for the supply anddemand of an agricultural product. In this example, a weather variable is used asan instrument to identify the demand schedule. That is, the identifyingrestriction is imposed on the model that weather does not affect the demand forthe agricultural good directly. Sims argues that even this assumption isundermined if one allows for expectations: “However certain we are that thetastes of consumers in the U.S. are unaffected by the temperature in Brazil, wemust admit that it is possible that U.S. consumers, upon reading of a frost inBrazil in the newspapers, might attempt to stockpile coffee in anticipation of thefrost’s effect on price. Thus variables known to affect supply enter the demandequation, and vice versa, through terms in expected price.”29The fact that identifying restrictions are often controversial can also beillustrated with the restrictions that have been imposed on the small structuralmodel considered here. Beginning with the identification of the money supplyrelation, it has been argued that the direct effect of fiscal policy on moneysupply can be restricted to zero on a priori grounds. Barro (1977) disagrees: Inan influential paper he argues that due to the seignorage to be gained fromexpanding the money supply there is an incentive for the government to fallback on this source of revenue when fiscal expenditure rises above trend.Accordingly he models the money supply in his model as a function of a fiscalpolicy proxy, while the effect of this variable on his aggregate demand variableis restricted to zero. Thus, Barro uses exactly the opposite identifying restrictionthan the one used here, where fiscal policy was assumed to be an importantfactor for demand fluctuations, but not for the monetary policy stance.The identifying restriction involving the term spread is also open to challenge.For the identification of the aggregate demand relation we assumed that the27 See Sims (1980), p. 6. For a similar argument see Bagliano and Favero, p. 1072.28 See Sims (1980), p. 4.29 Sims (1980), p. 6.

13spread does not enter this relation as a determining variable. However, in NewKeynesian models it is typically assumed that current real spending depends onthe expected future level of real spending. 30 Since the term spread is often usedas a predictor of future economic activity, one would expect this variable to havea direct effect on current aggregate demand, thereby invalidating the identifyingrestriction. 31Since the identifying restrictions used so far are vulnerable to criticism, thiswould suggest searching for another set of exogenous variables to help with theidentification of the aggregate demand and the money supply relations, but thechallenge to find a new set of exogenous variables returns the discussion to thefirst point stressed by Sims, namely that there are not so many credible exogenous variables to begin with. This example illustrates that it is quite hard to findsuitable instruments for identification in the traditional dynamic simultaneousequation approach.C.The SVAR methodologyI.The SVAR modelThe preceding discussion of the traditional approach to identification providesan useful background for the SVAR methodology. The bivariate structuralmodel introduced in the last section is used here as well to demonstrate theSVAR approach to identification. But before we can discuss this issue, we needto introduce the SVAR model itself. For this purpose is useful to rewrite thestr

Kiel Institute of World Economics Duesternbrooker Weg 120 24105 Kiel (Germany) Kiel Working Paper No. 1072 An Introduction into the SVAR Methodology: Identification, Interpretation and Limitations of SVAR models by Jan Gottschalk Preliminary, this version: August 2001 The responsibil